Coding theory applied to KU-algebras
By
Samy M.Mostafa a , Bayumy.A.Youssefb , Hussein A. Jad c
Abstract
The notion of a KU-valued function on a set is introduced and related properties are investigated. Codes
generated by KU-valued functions are established. Moreover, we will provide an algorithm which
allows us to find a KU-algebra starting from a given binary block code.
Keywords. KU-valued function, binary block code of KU-valued functions
AMS Classification. 06F35
Corresponding Author : Samy M. Mostafa ( [email protected] ) .
---------------------------------------------------------------------------------------------1. Introduction
BCK-algebras form an important class of logical algebras introduced by Iseki [5,6,7] and were
extensively investigated by several researchers. The class of all BCK-algebras is a quasivariety. Iseki
posed an interesting problem (solved by Wronski [13]) whether the class of BCK-algebras is a variety.
In connection with this problem, Komori [9] introduced a notion of BCC-algebras and Dudek [1]
redefined the notion of BCC-algebras by using a dual form of the ordinary definition in the sense of
Komori. Dudek and Zhang [2] introduced a new notion of ideals in BCC-algebras and described
connections between such ideals and congruences. C.Prabpayak and U.Leerawat ([11], [12]) introduced
a new algebraic structure which is called KU-algebra. They gave the concept of homomorphisms of KU
- algebras and investigated some related properties. These algebras form an important class of logical
algebras and have many applications to various domains of mathematics, such as, group theory,
functional analysis, fuzzy sets theory, probability theory, topology, etc. Coding theory is a very young
mathematical topic. It started on the basis of transferring information from one place to another. For
instance, suppose we are using electronic devices to transfer information (telephone, television, etc.).
Here, information is converted into bits of 1’s and 0’s and sent through a channel, for example a cable or
via satellite. Afterwards, the 1’s and 0’s are reconverted into information again. Due to technical
problems, one can assume that while the bits are sent through the channel, there is a positive probability
p that single bits are being changed. Thus the received bits could be wrong. The idea of coding theory is
to give a method of how to convert the information into bits, such that there are no mistakes in the
received information, or such that at least some of them are corrected. On this account, encoding and
decoding algorithms are used to convert and reconvert these bits properly. One of the recent applications
of BCK-algebras was given in the Coding theory [3,8 ,12]. In Coding Theory, a block code is an errorcorrecting code which encodes data in blocks. In the paper [8], the authors introduced the notion of
BCK-valued functions and investigate several properties. Moreover,they established block-codes by
using the notion of BCK-valued functions. they show that every finite BCK-algebra determines a blockcode constructed a finite binary block-codes associated to a finite BCK-algebra. In [3,12] provided an
algorithm which allows to find a BCK-algebra starting from a given binary block code.
In [12] the authors presented some new connections between BCK- algebras and binary block codes.
In this paper, we apply the code theory to KU- algebras and obtain some interesting results.
1
2. Preliminaries
Now, we will recall some known concepts related to KU-algebra from the literature which will be
helpful in further study of this article.
Definition2.1[10,11] Algebra(X, ∗, 0) of type (2, 0) is said to be a KU -algebra, if it satisfies the
following axioms:
( ku1 ) 0  x  x
( ku2 ) x  y  0  ( y  z )  ( x  z )  0, ( z * x)  ( z * y)  0
( ku3 ) x  ( y  z )  y  ( x  z )
( ku4 ) ( x  y )  [( y  z )  ( x  z )]  0 ,
Example 2.2 Let X = {0, 1, 2, 3, 4} in which * is defined by the following table
*
0
1
2
3
4
0
0
1
2
3
4
1
0
0
2
3
4
2
0
1
0
3
3
3
0
0
2
0
2
4
0
0
0
0
0
It is easy to show that X is KU-algebra.
In a KU-algebra, the following identities are true : If we put in (ku 4) y = x = 0 we get
(0 * 0)
*
[ (0 * z) * (0 * z) ] =0 and it follows that : (Ku5) z * z = 0 , if we put y = 0 in (ku4) , we get
(p1) z * (x * z ) = 0 .
A subset S of KU-algebra X is called sub-algebra of X if x * y  S, whenever x, y  S.
A non empty subset A of a KU-algebra X is called a KU-ideal of X if it satisfies the following
conditions:
(I1) 0  A,
(I2) x * (y * z)  A , y  A implies x * z  A , for all x , y , z  X .
Lemma 2.3 [9 ] In a KU-algebra (X, *, 0), the following hold:
x ≤ y imply y * z ≤ x * z .
Lemma 2.4 [10 ] If X is KU-algebra then y * [(y * x) * x] = 0.
2
3. KU-valued functions
In what follows let A and X denote a nonempty set and a KU-algebra respectively, unless otherwise
specified.
~
Definition 3.1 A mapping A : A  X is called a KU-valued function (briefly, KU-function) on A .
~
Definition 3.2 A cut function of A , for q  X is defined to be a mapping
~
~
~
Aq : A  {0,1} such that (x  A) Aq ( x)  1  A( x) * q  0 .
~
Obviously, Aq is the characteristic function of the following subset of A , called a cut subset or a q~ ~
cut of A : Aq ( x) :

~
x  A : A( x) * q  0 .

Example 3.3 Let A ={x, y, z} and let X = {0, a, b, c, d} is a KU-algebra with the following Cayley
table:
*
0
a
b
c
d
0
0
a
b
c
d
a
0
0
b
b
a
b
0
a
0
a
d
c
0
0
0
0
a
d
0
0
b
b
0
~
~ x
The function A : A  X given by A  
a
y z
 is a KU-function on A , and its cut subsets are
b c 
A0   , Aa  x , Ab  y , Ac  A , Ad  x
~
Proposition 3.4 Every KU-function A : A  X on A is represented by the infimum of the set
 q  X , Aq ( x)  1 , that is x  X : A~( x)  inf q  X , A~q ( x)  1 .


~
~
~
Proof. For any x  A . Let A( x)  q  X , then A( x) * q  0 and so Aq ( x)  1 q  X .
~
~
Assume that Ar ( x)  1 for r  X , then A ( x) * r  0  q * r , i.e r  q .
~
~
Since q  r  X , Ar ( x)  1 , for x  A , r  X ,it follows that A( x)  q  inf


This completes the proof.
3

~
r  X , Ar ( x)  1 .

~
Proposition 3.5 Let A : A  X be a KU-function on A . If q * p  0 for all p , q  X ,
we get Ap  Aq .
~
Proof. Let p , q  X , be such that q * p  0 and x  Ap , then A( x) * p  0
Using ( ku1 ) and ( ku2 ) ,we have
( KU 2 )



~
~
0  ( q * p ) * ( A( x) * P )  ( A( x) * q ) , and so x  Aq . Therefore Ap  Aq .
This completes the proof.
~
Proposition 3.6 Let A : A  X be KU-function on A . Then
~
~
1- (x, y  A)( A( x)  A( y )  AA~ ( x )  AA~ ( y )
~
2- (q  X )( x  A)( A( x) * q  0  AA~ ( x )  Aq
Proof. (1) The sufficiency is obvious. Assume that AA~ ( x )  AA~ ( y ) for all x, y  A .Then
AA~ ( y ) * AA~ ( x )  0 or AA~ ( x ) * AA~ ( y )  0 .Thus
AA~ ( x ) 
 z  A, A~( z) * A~( x)  0   z  A, A~( z ) * A~( y)  0  A
~
A( y)
(2) The necessity follows from Proposition 3.4. Let q  X and x  A be such that
~
~
~
AA~ ( x )  Aq . If A ( x) * q  0 then x  Aq . Since A ( x) * A ( x)  0 , it follows that x  AA~ ( x ) ,
so that AA~ ( x )  Aq .This is a contradiction.
~
Corollary 3.7 Let A : A  X be KU-function on A .Then
~
~
(x, y  A)( A( x) * A( y )  0  AA~ ( y )  AA~ ( x ) ) .
Proof. Straightforward.
~
For a KU-function A : A  X , consider the following sets:
~
~
Ax   Aq : q  X  ,
Ax  Aq : q  X .


~
Proposition 3.8 Let A : A  X be KU-function on A . Then
(Y  X )( inf Y in X  Ainf( q: qY )  Aq : q  Y .
Proof. Let (Y  X ) there exists inf Y in X such that x  Ainf( q: qY ) .We have
~
~
x  Ainf( q: qY )  A( x) * inf q : q  Y   0  (r  Y )( A( x) * r  0)  (r  Y )( x  Ar ) 
x  Aq : q  Y .This completes the proof.
4
~
Corollary 3.9 Let A : A  X be KU-function on A , where X is a bounded
KU-algebra, then S  X , Ainf( q: qS )  Aq : q  S  .
~
Corollary 3.10 Let A : A  X be KU-function on A , assume that for any Y  X
, there exists a infimum of Y such that ( p, q  Y ) , we have Ap  Aq  AX .
The following example shows that the converse of the corollary 3.10 may not true in general.
Example 3.11. Let A  x, y be a set and let X  0, a, b, c, d  be a KU-algebra with the following
Cayley table:
*
0
a
b
c
d
0
0
0
0
0
0
a
a
0
a
0
0
b
b
b
o
0
b
c
c
b
a
0
b
d
d
a
d
a
0
~
The function A : A  X given by
~ x y
 is a KU-function on A , then
A  
a b 
~
A0
~
Aa
~
Ab
~
Ac
~
Ad
x
a
0
y
b
0
1
0
0
1
1
1
1
0
And its cut subsets are
A0   , Aa  x , Ab  y
,
Ac  x, y
Note that Aa  Ab  x  y AX
,
Ad  x
, but inf a, b does exists in X .
5
~
Proposition 3.12 Let A : A  X be KU-function on A , then
 Aq q  X   A
~
Proof. Obviously,  Aq q  X   A .For every x  A , let A( x)  q  X .Then x  Aq and hence
x  Aq q  X  .Thus A  Aq q  X  .Therefore the result is valid.
~
Proposition 3.13 Let A : A  X be KU-function on A , then


(x  A)( Aq x  Aq  AX )
~
Proof. Note that for any x  A, x  Aq  Aq ( x)  1 ,
From Proposition 3.7 we get the following
~
 Aq x  Aq   Aq Aq ( x)  1  Ainf q A~ ( x ) 1  Aq . This completes the proof.

 

q
~
Let A : A  X be KU-function on A and  be a binary operation on X defined by
p, q  X ( pq  Ap  Aq ) . Then  is clearly an equivalence relation on X.


~
~
Let A( A)  q  X A( x)  q for some x  A and for q  X , ( q ]   x  X x * q  0
.
~
Proposition 3.14 For a KU-function A : A  X on A , we have
~
~
p, q  X ( pq  ( p]  A( A)  (q]  A( A)
Proof. We have pq  Ap  Aq
~
~
 (x  A) A( x) * p  0  A( x) * q  0
~
~
 x  A A( x)  ( p ]  x  A A( x)  (q ]
~
~
 ( p ]  A( A)  (q ]  A( A).



 

This completes the proof.
Example 3.15 Let X  an ; n  1,2,3,......,9 and define a binary operation  on X as follows
j
(ai , a j  X ) (ai  a j  ak ) , where k 
and (i, j ) is the least common divisor of i and j . Then
(i , j )
( X ;, ai ) is a KU-algebra. Its Cayley table is as follows:
6

a1
a2
a3
a4
a5
a6
a7
a8
a9
a1
a1
a2
a3
a4
a5
a6
a7
a8
a9
a2
a1
a1
a3
a2
a5
a3
a7
a4
a9
a3
a1
a2
a1
a4
a5
a2
a7
a8
a3
a4
a1
a1
a3
a1
a5
a3
a7
a2
a9
a5
a1
a2
a3
a4
a1
a6
a7
a8
a9
a6
a1
a1
a1
a2
a5
a1
a7
a4
a3
a7
a1
a2
a3
a4
a5
a6
a1
a8
a9
a8
a1
a1
a3
a1
a5
a3
a7
a1
a9
a9
a1
a2
a1
a4
a5
a2
a7
a8
a1
~
Let A  a, b, c, d , e and A : A  X be a KU-function defined by
~ a
A  
 a4
b
a6
c
a7
d
a1
e
a2

 . Then

*
~
Aa1
~
Aa2
~
Aa3
~
Aa4
~
Aa5
~
Aa6
~
Aa 7
~
Aa8
~
Aa9
a
a4
0
b
a6
0
c
a7
0
d
a1
1
e
a2
0
0
0
0
1
1
0
0
0
1
0
1
0
0
1
1
0
0
0
1
0
0
1
0
1
1
0
0
1
1
0
1
0
0
1
1
0
0
0
1
0
and cut sets of à are as follows:
~
~
~
~
~
~
~
~
~
Aa1  Aa3  Aa5  Aa9  d  , Aa2  d , e, Aa4  Aa8  a, d , e , Aa6  b, d , e, Aa7  c, d .
7
4. Codes generates by KU-functions
Let x

  y  A ; xy

; for any x  A , x

is called equivalence class containing x .
 
~
~
Lemma 4.1 Let A : A  X be a KU- function on A . For every x  A , we have A( x)  inf x , that is

~
A ( x) the least element of the  to which it belongs.
Proof. Straightforward.
Let A  1,2,3,....., n and X be a finite KU-algebra. Then every KU-function
~
A : A  X on A determines a binary block code V of length n in the following way: To every x
where x  A , there corresponds a codeword Vx  x1 x2 .....xn
Such that
~
xi  x j  Ax (i )  j for i  A and j  0,1.
Let Vx  x1 x2 .....xn , V y  y1 y2 .....yn be two code words belonging to a binary block-code V .
Define an order relation  c on the set of code words belonging to a binary block- code V as
follows: Vx  c Vy  xi  yi for i  1,2,...., n …… (4.1)
Example 4.2 Let X  0, a, b, c be a KU-algebra with the following Cayley table:
*
0
a
b
c
0
0
a
b
c
a
0
0
a
c
b
0
0
0
c
c
0
a
b
0
~
Let A : X  X be a KU-function on X given by
~ 0 a b c
 . Then
A  
0 a b c
~
Ax 0
~
A0 1
~
Aa 1
~
Ab 1
~
Ac 1
a
b
c
0
0
0
1
0
0
1
1
0
0
0
1
8

,
V  1000,1100,1110,1001. See Figure (1)
( X , )
(V , )
Generally, we have the following theorem.
Theorem 4.3 Every finite KU-algebra X determines a block-code V such that
( X , ) is isomorphic to ( X ,  c ) .
Proof. Let X  ai ; i  1,2,3,......, n be a finite KU-algebra in which a1 is the least element and let
~
~
~
A : X  X be identify KU-function on X .The decomposition of A provides a family Aq q  X which


is the desired code under the order Vx c V y  xi  yi for i  1,2,...., n
~
~
Let f : X  Aq ; q  X be a function defined by f (q)  Aq for all q  X .By lemma 4.1, every


 class contains exactly one element .So, f is one to one. Let x, y  X be such that y * x  a1 i.e x  y .
~
~
Then Ax  Ay (by Proposition 3.5), which means that Ax  Ay . Therefore f is an isomorphism.
This completes the proof.
Example 4.4
Consider a KU-algebra X  an ; n  1,2,3,......,9 which is considered in example 3.15.
~
Let A : X  X be a KU-function on X given by
~  a1
A  
 a1
a2
a3
a4
a5
a6
a7
a8
a2
a3
a4
a5
a6
a7
a8
a9 

a9 
9
Then
*
~
Aa1
~
Aa2
~
Aa3
~
Aa4
~
Aa5
~
Aa6
~
Aa 7
~
Aa8
~
Aa9
a1
1
a2
0
a3
0
a4
0
a5
0
a6
0
a7
0
a8
0
a9
0
1
1
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
1
1
0
1
0
0
0
0
0
1
0
0
0
1
0
0
0
0
1
1
1
0
0
1
0
0
0
1
0
0
0
0
0
1
0
0
1
1
0
1
0
0
0
1
0
1
0
1
0
0
0
0
0
1
Thus
V= {100000000, 110000000, 101000000, 110100000, 100010000, 11100100,100000100, 110100010,
101000001}. See Figure (2)
( X ,)
(V , )
References
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Inst. Math. Acad. Sinica, 20 (1992), 129-136.
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10
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,
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Samy M. Mostafa ([email protected])
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt.
Bayumy.A.Youssef ([email protected] )
Informatics Research Institute,City for Scientific Research and Technological Applications, Borg
ElArab, Alexandria, Egypt.
Hussein ali gad
([email protected] )
Department of Mathematics and Computer Science .Informatics Research Institute(IRI)
City of Scientific Research and Technological applications, Alexandria, Egypt
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